Consider the iterated function system $T_{1}(x)=(\beta x,\tau y)$, $T_{2}(x,y)=(\beta x+(1-\beta),\tau y+ (1-\tau))$ for $\beta\in(1/2,1)$ and $\tau\in (0,1/2)$ with self affine set $\Lambda_{\beta,\tau}$.

It is known that for almost all $\beta$ and all $\tau$ $\dim_{H}\Lambda_{\beta,\tau}=\dim b=1-\log(2\beta)/\log(\tau)$ where $b=(1/2,1/2)$ is the Bernoulli measure on $\Lambda$. Essential this is due to the fact that the projection of $b$ to the $x$-axis is an infinite Bernoulli convolution which is know to be generic absolutely continues (Solomyak theorem).

On the other hand we have shown that if $\beta^{-1}$ is a Pisot number we have $\dim b<\dim_{H}\Lambda_{\beta,\tau}<1-\log(2\beta)/\log(\tau)$ for all Bernoulli measures $b$. The projection of all $b$ in this case is singular with a dimension drop (Erdös theorem).

Now here comes the question: Is there a (ergodic) measure of full dimension and what is $\dim_{H}\Lambda_{\beta,\tau}$ in the later case?